JournalsrmiVol. 10, No. 3pp. 467–505

On singular integrals of Calderón-type in Rn\mathbb R^n, and BMO

  • Dorina Mitrea

    University of Missouri, Columbia, United States
On singular integrals of Calderón-type in $\mathbb R^n$, and BMO cover
Download PDF

Abstract

We prove LpL^p (and weighted LpL^p) bounds for singular integrals of the form

p.v.RnEA(x)A(y)xyΩ(xy)xynf(y)dy,\rm p.v. \int_{\mathbb R^n} E \lgroup \frac{A(x)–A(y)}{|x–y} \rgroup \frac{\Omega(x–y)}{|x–y|^n} f(y)dy,

where E(t)=E(t) = cos tt if Ω\Omega is odd, and E(t)=E(t) = sin tt if Ω\Omega is even, and where A\bigtriangledown A \in BMO. Even in the case that Ω\Omega is smooth, the theory of singular integrals with "rough" kernels plays a key role in the proof. By standard techniques, the trigonometric function EE can then be replaced by a large class of smooth functions FF. Some related operators are also considered. As a further application, we prove a compactness result for certain layer potentials.

Cite this article

Dorina Mitrea, On singular integrals of Calderón-type in Rn\mathbb R^n, and BMO. Rev. Mat. Iberoam. 10 (1994), no. 3, pp. 467–505

DOI 10.4171/RMI/159